Theorem ineccnvmo 38399

Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)

Assertion

Ref	Expression
ineccnvmo	⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)

Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem ineccnvmo

Step	Hyp	Ref	Expression
1		relcnv 6052	. . 3 ⊢ Rel ◡𝐹
2		id 22	. . . 4 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
3	2	inecmo 38397	. . 3 ⊢ (Rel ◡𝐹 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥))
4	1, 3	ax-mp 5	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥)
5		brcnvg 5818	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦))
6	5	el2v 3443	. . . 4 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
7	6	rmobii 3354	. . 3 ⊢ (∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
8	7	albii 1820	. 2 ⊢ (∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
9	4, 8	bitri 275	1 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)

Syntax hints:   ↔ wb 206   ∨ wo 847  ∀wal 1539   = wceq 1541  ∀wral 3047  ∃*wrmo 3345  Vcvv 3436   ∩ cin 3896  ∅c0 4280   class class class wbr 5089  ^◡ccnv 5613  Rel wrel 5619  [cec 8620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624

This theorem is referenced by:  ineccnvmo2  38402